Difference between revisions of "Dirichlet beta"
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(Created page with "$$\beta(x) = \displaystyle\sum_{k=0}^{\infty} (-1)^k (2k+1)^{-x} = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$ where $\Phi$ denotes the Lerch transcendent.") |
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| − | $$\beta( | + | __NOTOC__ |
| − | + | The Dirichlet $\beta$ function is defined by | |
| + | $$\beta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^z}.$$ | ||
| + | |||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Plot dirichlet beta.png|Graph of $\beta$ on $[-4,4]$. | ||
| + | File:Domain coloring dirichlet beta.png|[[Domain coloring]] of [[analytic continuation]] of $\beta$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | [[Catalan's constant using Dirichlet beta]]<br /> | ||
| + | [[Dirichlet beta in terms of Lerch transcendent]]<br /> | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 00:54, 11 December 2016
The Dirichlet $\beta$ function is defined by $$\beta(z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^z}.$$
Graph of $\beta$ on $[-4,4]$.
Domain coloring of analytic continuation of $\beta$.
Properties
Catalan's constant using Dirichlet beta
Dirichlet beta in terms of Lerch transcendent
